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The '''1600 equal divisions of the octave''' ('''1600edo'''), or the '''1600-tone equal temperament''' ('''1600tet'''), '''1600 equal temperament''' ('''1600et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1600 [[equal]] parts of exactly 750 [[cent|millicents]] each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
{{Harmonics in equal|1600}}
1600edo is a very strong 37-limit system, being [[consistency|distinctly consistent]] in the [[37-odd-limit]] with a smaller [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than anything else with this property until [[4501edo|4501]].
1600edo is a very strong 37-limit system, being distinctly consistent in the 37-limit with a smaller [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] than anything else with this property until [[4501edo|4501]]. It is also the first division past [[311edo|311]] with a lower 43-limit relative error. One step of it is the [[relative cent]] for [[16edo|16]].


1600's divisors are {{EDOs|1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, 800}}.
It is also the first division past [[311edo|311]] with a lower [[43-limit]] relative error, being ''almost'' consistent in the [[45-odd-limit]], missing only [[50/39]] and [[39/25]], both of which being off by ''52.6%'' by [[patent val]] mapping, which is still just an error of 0.3945 cents.
 
In the 7-limit, it supports [[crazy]], it supports In the 11-limit, it supports the rank-3 temperament [[thor]]. In higher limits, it tempers out [[4096/4095]] in the [[13-limit]] (allowing [[schisminic chords]]), [[12376/12375]] in the [[17-limit]], [[6860/6859]] in the 19-limit, and due to being consistent higher than 33-odd-limit it enables the essentially tempered [[flashmic chords]].
 
=== Prime harmonics ===
{{Harmonics in equal|1600|prec=3|columns=12}}{{Harmonics in equal|1600|columns=12|start=13|prec=3|collapsed=true|title=Approximation of prime harmonics in 1600edo (continued)}}
 
=== Subsets and supersets ===
Since 1600 factors into {{factorization|1600}}, 1600edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, and 800 }}.  
 
One step of it is the [[relative cent]] for [[16edo|16]]. Its high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for [[interval size measure]]. One step of 1600edo is already used as a measure called ''śata'' in the context of 16edo [[Armodue theory]]. Similar to the [[Mina]] in the [[27-odd-limit]], All [[45-odd limit]] intervals can be written using integer values of śata, being more in tune than out of tune.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| {{Monzo| -53 10 16 }}, {{monzo| 26 -75 40 }}
| {{Mapping| 1600 2536 3715 }}
| −0.0003
| 0.0228
| 3.04
|-
| 2.3.5.7
| 4375/4374, {{monzo| 36 -5 0 -10 }}, {{monzo| -17 5 16 -10 }}
| {{Mapping| 1600 2536 3715 4492 }}
| −0.0157
| 0.0332
| 4.43
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, {{monzo| 24 -1 -5 0 1 }}, {{monzo| 15 1 7 -8 -3 }}
| {{Mapping| 1600 2536 3715 4492 5535 }}
| −0.0172
| 0.0329
| 4.39
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543
| {{Mapping| 1600 2536 3715 4492 5535 5921 }}
| −0.0087
| 0.0356
| 4.75
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869
| {{Mapping| 1600 2536 3715 4492 5535 5921 6540 }}
| −0.0163
| 0.0331
| 4.41
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 2
| 217\1600
| 162.75
| 1125/1024
| [[Crazy]]
|-
| 32
| 23\1600
| 17.25
| ?
| [[Dam]] / [[dike]] / [[polder]]
|-
| 32
| 121\1600<br>(21/1600)
| 90.75<br>(15.75)
| 48828125/46294416<br>(?)
| [[Windrose]]
|-
| 32
| 357\1600<br>(7\1600)
| 267.75<br>(5.25)
| 245/143<br>(?)
| [[Germanium]]
|-
| 80
| 629\1600<br>(9\1600)
| 471.75<br>(6.75)
| 130/99<br>(?)
| [[Tetraicosic]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Latest revision as of 13:38, 13 March 2026

← 1599edo 1600edo 1601edo →
Prime factorization 26 × 52
Step size 0.75 ¢ 
Fifth 936\1600 (702 ¢) (→ 117\200)
Semitones (A1:m2) 152:120 (114 ¢ : 90 ¢)
Consistency limit 37
Distinct consistency limit 37

1600 equal divisions of the octave (abbreviated 1600edo or 1600ed2), also called 1600-tone equal temperament (1600tet) or 1600 equal temperament (1600et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1600 equal parts of exactly 0.75 ¢ each. Each step represents a frequency ratio of 21/1600, or the 1600th root of 2.

Theory

1600edo is a very strong 37-limit system, being distinctly consistent in the 37-odd-limit with a smaller relative error than anything else with this property until 4501.

It is also the first division past 311 with a lower 43-limit relative error, being almost consistent in the 45-odd-limit, missing only 50/39 and 39/25, both of which being off by 52.6% by patent val mapping, which is still just an error of 0.3945 cents.

In the 7-limit, it supports crazy, it supports In the 11-limit, it supports the rank-3 temperament thor. In higher limits, it tempers out 4096/4095 in the 13-limit (allowing schisminic chords), 12376/12375 in the 17-limit, 6860/6859 in the 19-limit, and due to being consistent higher than 33-odd-limit it enables the essentially tempered flashmic chords.

Prime harmonics

Approximation of prime harmonics in 1600edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.000 +0.045 -0.064 +0.174 -0.068 +0.222 +0.045 +0.237 +0.226 +0.173 +0.214 -0.094
Relative (%) +0.0 +6.0 -8.5 +23.2 -9.1 +29.6 +5.9 +31.6 +30.1 +23.0 +28.6 -12.5
Steps
(reduced) 		1600
(0) 		2536
(936) 		3715
(515) 		4492
(1292) 		5535
(735) 		5921
(1121) 		6540
(140) 		6797
(397) 		7238
(838) 		7773
(1373) 		7927
(1527) 		8335
(335)
Approximation of prime harmonics in 1600edo (continued)
Harmonic 41 43 47 53 59 61 67 71 73 79 83 89
Error Absolute (¢) -0.062 -0.018 -0.257 +0.245 -0.172 -0.135 +0.193 +0.303 +0.211 -0.037 -0.047 -0.130
Relative (%) -8.3 -2.4 -34.2 +32.7 -22.9 -18.0 +25.7 +40.5 +28.1 -4.9 -6.3 -17.3
Steps
(reduced) 		8572
(572) 		8682
(682) 		8887
(887) 		9165
(1165) 		9412
(1412) 		9489
(1489) 		9706
(106) 		9840
(240) 		9904
(304) 		10086
(486) 		10200
(600) 		10361
(761)

Subsets and supersets

Since 1600 factors into 26 × 52, 1600edo has subset edos 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, and 800.

One step of it is the relative cent for 16. Its high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for interval size measure. One step of 1600edo is already used as a measure called śata in the context of 16edo Armodue theory. Similar to the Mina in the 27-odd-limit, All 45-odd limit intervals can be written using integer values of śata, being more in tune than out of tune.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [-53 10 16⟩, [26 -75 40⟩ [⟨1600 2536 3715]] −0.0003 0.0228 3.04
2.3.5.7 4375/4374, [36 -5 0 -10⟩, [-17 5 16 -10⟩ [⟨1600 2536 3715 4492]] −0.0157 0.0332 4.43
2.3.5.7.11 3025/3024, 4375/4374, [24 -1 -5 0 1⟩, [15 1 7 -8 -3⟩ [⟨1600 2536 3715 4492 5535]] −0.0172 0.0329 4.39
2.3.5.7.11.13 3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543 [⟨1600 2536 3715 4492 5535 5921]] −0.0087 0.0356 4.75
2.3.5.7.11.13.17 2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869 [⟨1600 2536 3715 4492 5535 5921 6540]] −0.0163 0.0331 4.41

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
2 217\1600 162.75 1125/1024 Crazy
32 23\1600 17.25 ? Dam / dike / polder
32 121\1600
(21/1600) 		90.75
(15.75) 		48828125/46294416
(?) 		Windrose
32 357\1600
(7\1600) 		267.75
(5.25) 		245/143
(?) 		Germanium
80 629\1600
(9\1600) 		471.75
(6.75) 		130/99
(?) 		Tetraicosic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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